vector revision

# In 3d an orthogonal coordinate system is

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In 3D, an orthogonal coordinate system is characterised by ˆ ı · ˆ ı = ˆ · ˆ = ˆ k · ˆ k = 1 and ˆ ı · ˆ = ˆ · ˆ k = ˆ k · ˆ ı = 0

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A scalar product is an “inner product” 1.16 We have been writing vectors as row vectors a = [ a 1 , a 2 , a 3 ] It’s convenient: it takes less space than writing column vectors In matrix algebra, vectors are column vectors . So, M a = v means M 11 M 12 M 13 M 21 M 22 M 23 M 31 M 32 M 33 a 1 a 2 a 3 = v 1 v 2 v 3 and row vectors are written as a ( a transpose). Most times can be relaxed, but need to fuss to point out that the scalar product is also the inner product used in linear algebra. The inner product is defined as a b a b = [ a 1 , a 2 , a 3 ] b 1 b 2 b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3 = a · b
Scalar Product Example 1 1.17 Question A force F is applied to an object as it moves by a small amount δ r . What work is done on the object by the force? Answer The work done is equal to the component of force in the direction of the displacement multiplied by the displacement itself. This is just a scalar product: δW = F · δ r . Later we will see how to integrate such elements over particular paths as lineintegrals .

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Scalar Product Example 2 1.18 Question A cube has four diagonals, connecting opposite vertices. What is the angle between an adjacent pair? Answer k i j [-1,1,1] [1,1,1] The directions of the diagonals are [ ± 1 , ± 1 , ± 1]. The ones shown in the figure are [1 , 1 , 1] and [ 1 , 1 , 1]. The angle is thus θ = cos 1 [1 , 1 , 1] · [ 1 , 1 , 1] 1 2 + 1 2 + 1 2 1 2 + 1 2 + 1 2 = cos 1 (1 / 3)
Scalar Product Example 3 1.19 Question: Pinball with velocity s bounces (elastically) from a baffle whose endpoints are p and q . What is the velocity vector after the bounce? Answer ^ ^ p v u q s Refer to coord frame with principal directions along and perpendicular to the baffle: ˆu = [ u x , u y ] = q p | q p | ˆ v = u = [ u y , u x ] Before impact: velocity is s before = ( s . ˆu ) ˆu + ( s . ˆ v ) ˆ v After impact: component of velocity in dirn of baffle ˆu is same . component normal to the baffle along ˆ v is reversed s after = ( s . ˆu ) ˆu ( s . ˆ v ) ˆ v

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Scalar Product Example 3 1.20 Worth reflecting on this example ... Using vectors as complete entities (ie, not thinking about components) has made a tricky problem trivial to solve. Several languages (including Matlab) allow one to declare vector objects ^ ^ p v u q s p=[3;4] q=[1;-1] s=[1;2] diff=q-p uhat=diff/norm(diff) vhat=[-uhat(2);uhat(1)] safter=dot(s,uhat)*uhat-dot(s,vhat)*vhat You think in vectors, while built in routines handle the detail of components ... Reflection over.
Direction cosines use projection 1.21 The quantities λ = a · ˆ ı a , µ = a · ˆ a , ν = a · ˆ k a are the cosines of the angles which the vector a makes with the coordinate vectors ˆ ı , ˆ k They are the direction cosines of the vector a .

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